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Thinking of a Square...and a Circle

Note: While contemplating a square and a circle, these questions arose...have fun!

True or False: If a square's sides (or circle's circumference) are irrational, the diagonal (or radius) must be rational.

True or False: If a square's diagonal (or circle's radius) is irrational, the sides (circumference) must be rational.

Construction...With Fewest Steps: Construct a square (or circle) whose area is the sum of the areas of two different squares (circles).

Construction...With Fewest Steps: Construct a square (or circle) whose area is the difference of the areas of two different squares (circles).

Construction...With Fewest Steps: Construct a square whose area is the average of the areas of two different squares.

Construction...With Fewest Steps: Construct a square whose area is the geometric mean of the areas of two different squares.

Construction...With Fewest Steps: Construct a square whose area is the harmonic mean of the areas of two different squares.

Note: Do you think these same constructions could be accomplished (quickly!) if one was thinking of cubes (or spheres) rather than squares (circles)?

 


Hint: Draw pictures....and think of the Pythagorean Theorem.

Also, you will probably need to review construction techniques for the average, geometric mean, and harmonic means using line seqments.

 


Solution Commentary: No answers provided....as I am still playing with and extending this problem set.